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Properties

Label 2-525-105.32-c1-0-31
Degree $2$
Conductor $525$
Sign $0.424 - 0.905i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.15 + 0.578i)2-s + (1.42 + 0.988i)3-s + (2.59 + 1.5i)4-s + (2.5 + 2.95i)6-s + (−2.55 + 0.684i)7-s + (1.58 + 1.58i)8-s + (1.04 + 2.81i)9-s + (2.21 + 4.70i)12-s + (3.74 − 3.74i)13-s − 5.91·14-s + (−0.500 − 0.866i)16-s + (1.15 + 4.31i)17-s + (0.631 + 6.67i)18-s + (−1.73 + i)19-s + (−4.31 − 1.55i)21-s + ⋯
L(s)  = 1  + (1.52 + 0.409i)2-s + (0.821 + 0.570i)3-s + (1.29 + 0.750i)4-s + (1.02 + 1.20i)6-s + (−0.965 + 0.258i)7-s + (0.559 + 0.559i)8-s + (0.348 + 0.937i)9-s + (0.638 + 1.35i)12-s + (1.03 − 1.03i)13-s − 1.58·14-s + (−0.125 − 0.216i)16-s + (0.280 + 1.04i)17-s + (0.148 + 1.57i)18-s + (−0.397 + 0.229i)19-s + (−0.940 − 0.338i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.424 - 0.905i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.21798 + 2.04434i\)
\(L(\frac12)\) \(\approx\) \(3.21798 + 2.04434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 - 0.988i)T \)
5 \( 1 \)
7 \( 1 + (2.55 - 0.684i)T \)
good2 \( 1 + (-2.15 - 0.578i)T + (1.73 + i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.74 + 3.74i)T - 13iT^{2} \)
17 \( 1 + (-1.15 - 4.31i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.73 - i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + 6.47i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.91T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.73 + 10.2i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.91iT - 41T^{2} \)
43 \( 1 + (-1.87 + 1.87i)T - 43iT^{2} \)
47 \( 1 + (8.63 + 2.31i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.31 - 1.15i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.91 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.55 + 0.684i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (-1.36 - 5.11i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.73 - i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.58 - 1.58i)T + 83iT^{2} \)
89 \( 1 + (-2.95 - 5.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.48 - 7.48i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.96311404049989077436472626826, −10.29474444737980100852385344591, −9.135724476825211992947684602601, −8.306884911281049680953235780515, −7.18114896143532106417804459433, −6.08452195608322156553824031942, −5.44610487845166151817795816814, −4.06461888739328673096675745444, −3.54580543787896586027491166206, −2.51303076505837124801440893747, 1.70384509049486757411128774348, 3.09197897751909378676358303881, 3.61975774153974585142957268116, 4.77089367645210767263153015489, 6.18699874080556163481629300164, 6.67629349805660124735908025748, 7.79426988592510927477015284086, 9.118331568172178225130002764141, 9.707067653649457631996843788607, 11.23385337024712598638626105923

Graph of the $Z$-function along the critical line